Alternatively, a regular language can be defined as a language recognized by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem.[3] In the Chomsky hierarchy, regular languages are defined to be the languages that are generated by Type-3 grammars (regular grammars).

All finite languages are regular; in particular the empty string language {ε} = Ø* is regular. Other typical examples include the language consisting of all strings over the alphabet {a, b} which contain an even number of as, or the language consisting of all strings of the form: several as followed by several bs.

A simple example of a language that is not regular is the set of strings .[4] Intuitively, it cannot be recognized with a finite automaton, since a finite automaton has finite memory and it cannot remember the exact number of a's. Techniques to prove this fact rigorously are given below.

it is recognized by some finite monoid, meaning it is the preimage of a subset of a finite monoid under a homomorphism from the free monoid on its alphabet[note 7]

Some authors use one of the above properties different from "1." as alternative definition of regular languages.

Some of the equivalences above, particularly those among the first four formalisms, are called Kleene's theorem in textbooks. Precisely which one (or which subset) is called such varies between authors. One textbook calls the equivalence of regular expressions and NFAs ("1." and "2." above) "Kleene's theorem".[6] Another textbook calls the equivalence of regular expressions and DFAs ("1." and "3." above) "Kleene's theorem".[7] Two other textbooks first prove the expressive equivalence of NFAs and DFAs ("2." and "3.") and then state "Kleene's theorem" as the equivalence between regular expressions and finite automata (the latter said to describe "recognizable languages").[2][8] A linguistically oriented text first equates regular grammars ("4." above) with DFAs and NFAs, calls the languages generated by (any of) these "regular", after which it introduces regular expressions which it terms to describe "rational languages", and finally states "Kleene's theorem" as the coincidence of regular and rational languages.[9] Other authors simply define "rational expression" and "regular expressions" as synonymous and do the same with "rational languages" and "regular languages".[1][2]

the trio operations: string homomorphism, inverse string homomorphism, and intersection with regular languages. As a consequence they are closed under arbitrary finite state transductions, like quotient with a regular language. Even more, regular languages are closed under quotients with arbitrary languages: If L is regular then L/K is regular for any K.

To locate the regular languages in the Chomsky hierarchy, one notices that every regular language is context-free. The converse is not true: for example the language consisting of all strings having the same number of a's as b's is context-free but not regular. To prove that a language such as this is not regular, one often uses the Myhill–Nerode theorem or the pumping lemma among other methods.[12]

There are two purely algebraic approaches to define regular languages. If:

The language L is regular if and only if the number of equivalence classes of ~ is finite (A proof of this is provided in the article on the syntactic monoid). When a language is regular, then the number of equivalence classes is equal to the number of states of the minimal deterministic finite automaton accepting L.

A similar set of statements can be formulated for a monoid . In this case, equivalence over M leads to the concept of a recognizable language.

In computational complexity theory, the complexity class of all regular languages is sometimes referred to as REGULAR or REG and equals DSPACE(O(1)), the decision problems that can be solved in constant space (the space used is independent of the input size). REGULAR ≠ AC0, since it (trivially) contains the parity problem of determining whether the number of 1 bits in the input is even or odd and this problem is not in AC0.[13] On the other hand, REGULAR does not contain AC0, because the nonregular language of palindromes, or the nonregular language can both be recognized in AC0.[14]

If a language is not regular, it requires a machine with at least Ω(log log n) space to recognize (where n is the input size).[15] In other words, DSPACE(o(log log n)) equals the class of regular languages. In practice, most nonregular problems are solved by machines taking at least logarithmic space.

The generating function of a language L is a rational function if L is regular.[18] Hence for any regular language there exist an integer constant , complex constants and complex polynomials such that for every the number of words of length in is .[20][21][22][23]

Thus, non-regularity of certain languages can be proved by counting the words of a given length in . Consider, for example, the Dyck language of strings of balanced parentheses. The number of words of length in the Dyck language is equal to the Catalan number, which is not of the form , witnessing the non-regularity of the Dyck language. Care must be taken since some of the eigenvalues could have the same magnitude. For example, the number of words of length in the language of all even binary words is not of the form , but the number of words of even or odd length are of this form; the corresponding eigenvalues are . In general, for every regular language there exists a constant such that for all , the number of words of length is asymptotically .[24]

The notion of a regular language has been generalized to infinite words (see ω-automata) and to trees (see tree automaton).

Rational set generalizes the notion (of regular/rational language) to monoids that are not necessarily free. Likewise, the notion of a recognizable language (by a finite automaton) has namesake as recognizable set over a monoid that is not necessarily free. Howard Straubing notes in relation to these facts that “The term "regular language" is a bit unfortunate. Papers influenced by Eilenberg's monograph[27] often use either the term "recognizable language", which refers to the behavior of automata, or "rational language", which refers to important analogies between regular expressions and rational power series. (In fact, Eilenberg defines rational and recognizable subsets of arbitrary monoids; the two notions do not, in general, coincide.) This terminology, while better motivated, never really caught on, and "regular language" is used almost universally.”[28]

Each category of languages, except those marked by a *, is a proper subset of the category directly above it.Any language in each category is generated by a grammar and by an automaton in the category in the same line.